We introduce a new type of Gutzwiller variational wave function for correlated electrons coupled to phonons, able to treat on equal footing electronic and lattice degrees of freedom. We benchmark the wave function in the infinite-U Hubbard-Holstein model away from half-filling on a Bethe lattice, where we can directly compare with exact results by the Dynamical Mean-Field Theory. For this model, we find that variational results agree perfectly well with the exact ones. In particular the wave function correctly describes the crossover to a heavy polaron gas upon increasing the electron-phonon coupling....more